20.5: Uses of Radioisotopes
-
List common applications of radioactive isotopes
Radioactive isotopes have the same chemical properties as stable isotopes of the same element, but they emit radiation, which can be detected. If we replace one (or more) atom(s) with radioisotope(s) in a compound, we can track them by monitoring their radioactive emissions. This type of compound is called a radioactive tracer (or radioactive label ). Radioisotopes are used to follow the paths of biochemical reactions or to determine how a substance is distributed within an organism. Radioactive tracers are also used in many medical applications, including both diagnosis and treatment. They are used to measure engine wear, analyze the geological formation around oil wells, and much more.
Radioisotopes have revolutionized medical practice, where they are used extensively. Over 10 million nuclear medicine procedures and more than 100 million nuclear medicine tests are performed annually in the United States. Four typical examples of radioactive tracers used in medicine are technetium-99 \(\ce{(^{99}_{43}Tc)}\), thallium-201 \(\ce{(^{201}_{81}Tl)}\), iodine-131 \(\ce{(^{131}_{53}I)}\), and sodium-24 \(\ce{(^{24}_{11}Na)}\). Damaged tissues in the heart, liver, and lungs absorb certain compounds of technetium-99 preferentially. After it is injected, the location of the technetium compound, and hence the damaged tissue, can be determined by detecting the γ rays emitted by the Tc-99 isotope. Thallium-201 (Figure \(\PageIndex{1}\)) becomes concentrated in healthy heart tissue, so the two isotopes, Tc-99 and Tl-201, are used together to study heart tissue. Iodine-131 concentrates in the thyroid gland, the liver, and some parts of the brain. It can therefore be used to monitor goiter and treat thyroid conditions, such as Grave’s disease, as well as liver and brain tumors. Salt solutions containing compounds of sodium-24 are injected into the bloodstream to help locate obstructions to the flow of blood.
Radioisotopes used in medicine typically have short half-lives—for example, the ubiquitous Tc-99m has a half-life of 6.01 hours. This makes Tc-99m essentially impossible to store and prohibitively expensive to transport, so it is made on-site instead. Hospitals and other medical facilities use Mo-99 (which is primarily extracted from U-235 fission products) to generate Tc-99. Mo-99 undergoes β decay with a half-life of 66 hours, and the Tc-99 is then chemically extracted (Figure \(\PageIndex{2}\)). The parent nuclide Mo-99 is part of a molybdate ion, \(\ce{MoO4^2-}\); when it decays, it forms the pertechnetate ion, \(\ce{TcO4-}\). These two water-soluble ions are separated by column chromatography, with the higher charge molybdate ion adsorbing onto the alumina in the column, and the lower charge pertechnetate ion passing through the column in the solution. A few micrograms of Mo-99 can produce enough Tc-99 to perform as many as 10,000 tests.
Radioisotopes can also be used, typically in higher doses than as a tracer, as treatment. Radiation therapy is the use of high-energy radiation to damage the DNA of cancer cells, which kills them or keeps them from dividing (Figure \(\PageIndex{3}\)). A cancer patient may receive external beam radiation therapy delivered by a machine outside the body, or internal radiation therapy (brachytherapy) from a radioactive substance that has been introduced into the body. Note that chemotherapy is similar to internal radiation therapy in that the cancer treatment is injected into the body, but differs in that chemotherapy uses chemical rather than radioactive substances to kill the cancer cells.
Cobalt-60 is a synthetic radioisotope produced by the neutron activation of Co-59, which then undergoes β decay to form Ni-60, along with the emission of γ radiation. The overall process is:
\[\ce{^{59}_{27}Co + ^1_0n⟶ ^{60}_{27}Co⟶ ^{60}_{28}Ni + ^0_{−1}β + 2^0_0γ} \nonumber \]
The overall decay scheme for this is shown graphically in Figure \(\PageIndex{4}\).
Radioisotopes are used in diverse ways to study the mechanisms of chemical reactions in plants and animals. These include labeling fertilizers in studies of nutrient uptake by plants and crop growth, investigations of digestive and milk-producing processes in cows, and studies on the growth and metabolism of animals and plants.
For example, the radioisotope C-14 was used to elucidate the details of how photosynthesis occurs. The overall reaction is:
\[\ce{6CO2}(g)+\ce{6H2O}(l)⟶\ce{C6H12O6}(s)+\ce{6O2}(g), \nonumber \]
but the process is much more complex, proceeding through a series of steps in which various organic compounds are produced. In studies of the pathway of this reaction, plants were exposed to CO 2 containing a high concentration of \(\ce{^{14}_6C}\). At regular intervals, the plants were analyzed to determine which organic compounds contained carbon-14 and how much of each compound was present. From the time sequence in which the compounds appeared and the amount of each present at given time intervals, scientists learned more about the pathway of the reaction.
Commercial applications of radioactive materials are equally diverse (Figure \(\PageIndex{5}\)). They include determining the thickness of films and thin metal sheets by exploiting the penetration power of various types of radiation. Flaws in metals used for structural purposes can be detected using high-energy gamma rays from cobalt-60 in a fashion similar to the way X-rays are used to examine the human body. In one form of pest control, flies are controlled by sterilizing male flies with γ radiation so that females breeding with them do not produce offspring. Many foods are preserved by radiation that kills microorganisms that cause the foods to spoil.
Americium-241, an α emitter with a half-life of 458 years, is used in tiny amounts in ionization-type smoke detectors (Figure \(\PageIndex{6}\)). The α emissions from Am-241 ionize the air between two electrode plates in the ionizing chamber. A battery supplies a potential that causes movement of the ions, thus creating a small electric current. When smoke enters the chamber, the movement of the ions is impeded, reducing the conductivity of the air. This causes a marked drop in the current, triggering an alarm.
Summary
Compounds known as radioactive tracers can be used to follow reactions, track the distribution of a substance, diagnose and treat medical conditions, and much more. Other radioactive substances are helpful for controlling pests, visualizing structures, providing fire warnings, and for many other applications. Hundreds of millions of nuclear medicine tests and procedures, using a wide variety of radioisotopes with relatively short half-lives, are performed every year in the US. Most of these radioisotopes have relatively short half-lives; some are short enough that the radioisotope must be made on-site at medical facilities.  The radiation used for this treatment may be delivered externally or internally.
Glossary
- chemotherapy
- similar to internal radiation therapy, but chemical rather than radioactive substances are introduced into the body to kill cancer cells
- external beam radiation therapy
- radiation delivered by a machine outside the body
- internal radiation therapy
- (also, brachytherapy) radiation from a radioactive substance introduced into the body to kill cancer cells
- radiation therapy
- use of high-energy radiation to damage the DNA of cancer cells, which kills them or keeps them from dividing
- radioactive tracer
- (also, radioactive label) radioisotope used to track or follow a substance by monitoring its radioactive emissions


13.1: Mobility Introduction
- Assess factors that put patients at risk for problems with mobility
- Identify factors related to mobility across the life span
- Assess the effects of immobility on body systems
- Detail the nursing measures to prevent complications of immobility
- Promote the use of effective techniques of body mechanics among caregivers, patients, and significant others
- Identify evidence-based practices
Sit on a sturdy chair with your legs and arms stretched out in front of you, and then try to stand. This basic mobility task can be impaired during recovery from major surgery or for patients with chronic musculoskeletal conditions. Mobility, which includes moving one’s extremities, changing positions, sitting, standing, and walking, helps avoid degradation of many body systems and prevents complications associated with immobility. Nurses assist patients to be as mobile as possible, based on their individual circumstances, to achieve their highest level of independence, prevent complications, and promote a feeling of well-being. This chapter will discuss nursing assessments and interventions related to promoting mobility.


10.1.4: References, Key Terms and Concepts
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Bruhn, John G. and Howard M. Rebach. 2007. Sociological Practice: Intervention and Social Change . 2nd ed. New York, NY: Springer.
Lindblom, Charles E. and David K. Cohen. 1979. Usable Knowledge: Social Science and Social Problem Solving . New Have: Yale University Press.
Steele, Stephen F. and Jammie Price. 2008. Applied Sociology: Terms, Topics, Tools, and Tasks . 2nd ed. Belmont, CA: Thomson Wadsworth.
Viola, Judah J. and Susan Dvorak McMahon. 2010. Consulting and Evaluation with Nonprofit and Community-based Organizations . Sudbury, MA: Jones and Bartlett Publishers.
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Assessment Baseline data Clinical approach Community based approach Comprehensive intervention Continual improvement Evaluation Formative evaluation Human ecology approach Implementation Indicated level of intervention |
Life cycle approach Outcome objectives Plan Process objectives Process of intervention Role-conflict Selective level of intervention Social norms approach Social systems approach Summative evaluation Universal level of intervention |


9.5: Summary
The topics covered in this chapter can be summarized as follows:
| Section | Summary |
|---|---|
| 9.1 Understanding Earth Through Seismology | Seismic waves that travel through Earth are either P-waves (compression, or “push” waves) or S-waves (shear waves). P-waves are faster than S-waves, and can pass through fluids. By studying seismic waves, we can discover the nature and temperature characteristics of the various parts of Earth’s interior. |
| 9.2 The Temperature of Earth’s Interior | Earth’s temperature increases with depth (to around 5000°C at the centre), but there are significant variations in the rate of temperature increase. These variations are related to differences in composition and the existence of convection in the mantle and liquid part of the core. |
| 9.3 Earth’s Magnetic Field | Because of outer-core convection, Earth has a magnetic field. The magnetic force directions are different at different latitudes. The polarity of the field is not constant, and has flipped from “normal” (as it is now) to reversed and back to normal hundreds of times in the past. |
| 9.4 Isostasy | The “plastic” nature of the mantle, which allows for mantle convection, also determines the nature of the relationship between the crust and the mantle. The crust floats on the mantle in an isostatic relationship. Where the crust becomes thicker because of mountain building, it pushes farther down into the mantle. Oceanic crust, being heavier than continental crust, floats lower on the mantle. |
Answers to Review Questions can be found in Appendix 2 .
- What parts of Earth are most closely represented by typical stony meteorites and typical iron meteorites?
- On the below diagram draw (from memory) and label the approximate locations of the following boundaries: crust/mantle, mantle/core, outer core/inner core.
- Describe the important differences between P-waves and S-waves.
- Why does P-wave velocity decrease dramatically at the core-mantle boundary?
- Why do both P-waves and S-waves gradually bend as they move through the mantle?
- What is the evidence for mantle convection, and what is the mechanism that causes it?
- Where and how is Earth’s magnetic field generated?
- When were the last two reversals of Earth’s magnetic field?
- What property of the mantle is essential for the isostatic relationship between the crust and the mantle?
- How would you expect the depth to the crust-mantle boundary in the area of the Rocky Mountains to differ from that in central Saskatchewan?
- As you can see in Figure 9.22, British Columbia is still experiencing weak post-glacial isostatic uplift, especially in the interior, but also along the coast. Meanwhile offshore areas are experiencing weak isostatic subsidence. Why?


3.6E: Exercises
Practice Makes Perfect
Find the Domain and Range of a Relation
In the following exercises, for each relation a. find the domain of the relation b. find the range of the relation.
1. \({\{(1,4),(2,8),(3,12),(4,16),(5,20)}\}\)
- Answer
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a. \({\{1, 2, 3, 4, 5}\}\) b. \({\{4, 8, 12, 16, 20}\}\)
2. \({\{(1,−2),(2,−4),(3,−6),(4,−8),(5,−10)}\}\)
3. \({\{(1,7),(5,3),(7,9),(−2,−3),(−2,8)}\}\)
- Answer
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a. \({\{1, 5, 7, −2}\}\) b. \({\{7, 3, 9, −3, 8}\}\)
4. \({\{(11,3),(−2,−7),(4,−8),(4,17),(−6,9)}\}\)
In the following exercises, use the mapping of the relation to a. list the ordered pairs of the relation, b. find the domain of the relation, and c. find the range of the relation.
5.
- Answer
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a. (Rebecca, January 18), (Jennifer, April 1), (John, January 18), (Hector, June 23), (Luis, February 15), (Ebony, April 7), (Raphael, November 6), (Meredith, August 19), (Karen, August 19), (Joseph, July 30)
b. {Rebecca, Jennifer, John, Hector, Luis, Ebony, Raphael, Meredith, Karen, Joseph}
c. {January 18, April 1, June 23, February 15, April 7, November 6, August 19, July 30}
6.
7. For a woman of height \(5'4''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5–24.9\) is considered healthy.
- Answer
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a. \((+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5)\) b. \({\{+100, 110, 120, 130, 140, 150, 160,}\}\) c. \({\{17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}\}\)
8. For a man of height \(5'11''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5–24.9\) is considered healthy.
In the following exercises, use the graph of the relation to a. list the ordered pairs of the relation b. find the domain of the relation c. find the range of the relation.
9.
- Answer
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a. \((2, 3), (4, −3), (−2, −1), (−3, 4), (4, −1), (0, −3)\) b. \({\{−3, −2, 0, 2, 4}\}\)
c. \({\{−3, −1, 3, 4}\}\)
10.
11.
- Answer
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a. \((1, 4), (1, −4), (−1, 4), (−1, −4), (0, 3), (0, −3)\) b. \({\{−1, 0, 1}\}\) c. \({\{−4, −3, 3,4}\}\)
12.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to a. determine whether the relation is a function, b. find the domain of the relation, and c. find the range of the relation.
13. \( {\{(−3,9),(−2,4),(−1,1), (0,0),(1,1),(2,4),(3,9)}\}\)
- Answer
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a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{9, 4, 1, 0}\}\)
14. \({\{(9,−3),(4,−2),(1,−1),(0,0),(1,1),(4,2),(9,3)}\}\)
15. \({\{(−3,27),(−2,8),(−1,1), (0,0),(1,1),(2,8),(3,27)}\}\)
- Answer
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a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{0, 1, 8, 27}\}\)
16. \({\{(−3,−27),(−2,−8),(−1,−1), (0,0),(1,1),(2,8),(3,27)}\}\)
In the following exercises, use the mapping to a. determine whether the relation is a function, b. find the domain of the function, and c. find the range of the function.
17.
- Answer
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a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{0, 1, 2, 3}\}\)
18.
19.
- Answer
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a. no b. {Jenny, R and y, Dennis, Emily, Raul} c. {RHern and<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, R and<EMAIL_ADDRESS>}
20.
In the following exercises, determine whether each equation is a function.
21. a. \(2x+y=−3\)
b. \(y=x^2\)
c. \(x+y^2=−5\)
- Answer
-
a. yes b. yes c. no
22. a. \(y=3x−5\)
b. \(y=x^3\)
c. \(2x+y^2=4\)
23. a. \(y−3x^3=2\)
b. \(x+y^2=3\)
c. \(3x−2y=6\)
- Answer
-
a. yes b. no c. yes
24. a. \(2x−4y=8\)
b. \(−4=x^2−y\)
c. \(y^2=−x+5\)
Find the Value of a Function
In the following exercises, evaluate the function: a. \(f(2)\) b. \(f(−1)\) c. \(f(a)\).
25. \(f(x)=5x−3\)
- Answer
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a. \(f(2)=7\) b. \(f(−1)=−8\) c. \(f(a)=5a−3\)
26. \(f(x)=3x+4\)
27. \(f(x)=−4x+2\)
- Answer
-
a. \(f(2)=−6\) b. \(f(−1)=6\) c. \(f(a)=−4a+2\)
28. \(f(x)=−6x−3\)
29. \(f(x)=x^2−x+3\)
- Answer
-
a. \(f(2)=5\) b. \(f(−1)=5\)
c. \(f(a)=a^2−a+3\)
30. \(f(x)=x^2+x−2\)
31. \(f(x)=2x^2−x+3\)
- Answer
-
a. \(f(2)=9\) b. \(f(−1)=6\)
c. \(f(a)=2a^2−a+3\)
32. \(f(x)=3x^2+x−2\)
In the following exercises, evaluate the function: a. \(g(h^2)\) b. \(g(x+2)\) c. \(g(x)+g(2)\).
33. \(g(x)=2x+1\)
- Answer
-
a. \(g(h^2)=2h^2+1\)
b. \(g(x+2)=4x+5\)
c. \(g(x)+g(2)=2x+6\)
34. \(g(x)=5x−8\)
35. \(g(x)=−3x−2\)
- Answer
-
a. \(g(h^2)=−3h^2−2\)
b. \(g(x+2)=−3x−8\)
c. \(g(x)+g(2)=−3x−10\)
36. \(g(x)=−8x+2\)
37. \(g(x)=3−x\)
- Answer
-
a. \(g(h^2)=3−h^2\)
b. \(g(x+2)=1−x\)
c. \(g(x)+g(2)=4−x\)
38. \(g(x)=7−5x\)
In the following exercises, evaluate the function.
39. \(f(x)=3x^2−5x\); \(f(2)\)
- Answer
-
2
40. \(g(x)=4x^2−3x\); \(g(3)\)
41. \(F(x)=2x^2−3x+1\); \(F(−1)\)
- Answer
-
6
42. \(G(x)=3x^2−5x+2\); \(G(−2)\)
43. \(h(t)=2|t−5|+4\); \(f(−4)\)
- Answer
-
22
44. \(h(y)=3|y−1|−3\); \(h(−4)\)
45. \(f(x)=x+2x−1\); \(f(2)\)
- Answer
-
4
46. \(g(x)=x−2x+2\); \(g(4)\)
In the following exercises, solve.
47. The number of unwatched shows in Sylvia’s DVR is 85. This number grows by 20 unwatched shows per week. The function \(N(t)=85+20t\) represents the relation between the number of unwatched shows, N , and the time, t , measured in weeks.
a. Determine the independent and dependent variable.
b. Find \(N(4)\). Explain what this result means
- Answer
-
a. t IND; N DEP
b. \(N(4)=165\) the number of unwatched shows in Sylvia’s DVR at the fourth week.
48. Every day a new puzzle is downloaded into Ken’s account. Right now he has 43 puzzles in his account. The function \(N(t)=43+t\) represents the relation between the number of puzzles, N , and the time, t , measured in days.
a. Determine the independent and dependent variable.
b. Find \(N(30)\). Explain what this result means.
49. The daily cost to the printing company to print a book is modeled by the function \(C(x)=3.25x+1500\) where C is the total daily cost and x is the number of books printed.
a. Determine the independent and dependent variable.
b. Find \(N(0)\). Explain what this result means.
c. Find \(N(1000)\). Explain what this result means.
- Answer
-
a. x IND; C DEP
b. \(N(0)=1500\) the daily cost if no books are printed
c. \(N(1000)=4750\) the daily cost of printing 1000 books
50. The daily cost to the manufacturing company is modeled by the function \(C(x)=7.25x+2500\) where \(C(x)\) is the total daily cost and x is the number of items manufactured.
a. Determine the independent and dependent variable.
b. Find \(C(0)\). Explain what this result means.
c. Find \(C(1000)\). Explain what this result means.
Writing Exercises
51. In your own words, explain the difference between a relation and a function.
52. In your own words, explain what is meant by domain and range.
53. Is every relation a function? Is every function a relation?
54. How do you find the value of a function?
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


25.6: Parent-Newborn Bonding and Attachment
By the end of this section, you will be able to:
- Identify behaviors that show emotional attachment between the newborn and family
- Describe nursing interventions that support attachment and bonding
- Describe sibling adjustment and grandparent adaptation to the newborn
Assessing attachment behaviors in parents or caregivers and their newborn requires skills beyond those required for a physical assessment. Nurses rely on their interviewing skills and keen observation of the interactions between the caregivers and their newborn. Bowlby introduced attachment theory in the 1950s and over the following decades worked with Ainsworth to further flesh out the theory over many different disciplines (Bowlby, 1979). Bowlby wrote, “The propensity to make strong emotional bonds to particular individuals is a basic component of human nature.”
Attachment or the lack of it has serious long-lasting effects on the child’s development and relationships throughout life, particularly with their significant other and with their own children. Having a caregiver who is responsive and present for the infant gives the infant a foundation to make good relationships in the future, to have confidence to explore their environment, and to have strong self-esteem (Cornell & Drew, 2022).
A positive, robust primary caregiver and infant relationship is paramount to healthy growth and development. For example, bonding results from the infant and caregiver having experiences they enjoy together. Also important for the relationship is attachment , the more integral provision of a secure environment for the infant throughout their progression and exploration (Winston & Chicot, 2016). A number of factors can affect bonding and attachment postdelivery. For example, the high-risk infant admitted to an ICU, NICU, PICU, or CICU may have anomalies to the face or body. The child and caregiver may be unable to bond during direct breast-feeding or direct skin-to-skin contact because of the infant’s acute instability (Kim et al., 2020). Advances in medical technology allow special needs infants to survive more often than they did in the past. However, impaired attachment in the hospital related to the alteration of the parental role, posttraumatic shock of families, and potentially neurodevelopmental disabilities of the infant continue to be an issue in their care (Kim et al., 2020).
Behaviors That Show Attachment
Attachment is assessed by the nurse as they assess the family and the newborn. Confirmation of strong attachment between the birthing person and infant includes multiple cues:
- A visually alert infant makes eye contact with their birthing person, tracking and following the parent with their eyes.
- The parent-infant dyad smile at one another.
- The infant cries only when hungry or wet.
- The parent anticipates feedings by reading the infant’s feeding cues.
- The infant enjoys being cuddled, clings to the parent, and is easily consolable. (Wittkowski et al., 2020)
Concerning actions or behaviors from the infant include crying for hours, appearing inconsolable or colicky, engaging in unpredictable feeding or sleeping patterns, showing no preference for parents over others, infrequently smiling or having a bland facial expression, and resisting being held or cuddled.
Four Stages of Attachment
The four stages of attachment derive from a psychologic study by Schaffer and Emerson in the 1960s. Sixty infant subjects, all living in Glasgow, Scotland, were visited for developmental assessment at monthly intervals for the first 18 months of life (Schaffer & Emerson, 1964). The study found that the caregiver who had the infant’s attachment was the one who was the most sensitive and responsive to the infant's signals. The development of attachment begins with the asocial stage . This stage lasts from 0 to 6 weeks, and infants in it display a general lack of attachment. In the indiscriminate stage , the 6-week-old to 6-month-old infant is interested in others but consolable by all. Between 6 and 10 months is the specific stage , in which usually only one person is able to console the infant. The infant older than 10 months has many attachments and people who can console them; they are in the multiple stage . This mirrors the stages Bowlby found in the late 1950s. Table 25.8 provides more information about the stages of attachment (Tutor2U, 2021).
| Name of Stage | Timeframe | Description |
|---|---|---|
| Asocial or preattachment | 0–6 weeks |
Similar responses to objects and people
Preference for faces/eyes |
| Indiscriminate or attachment in making | 6 weeks to 6 months |
Preference for human company
Ability to distinguish between people but comforted indiscriminately |
| Specific or clear-cut attachment | 6–8 months to 18–24 months |
Infants show a preference for one caregiver, displaying separation and stranger anxiety
The infant looks to particular people for security, comfort, and protection |
| Multiple or formation of reciprocal relationship | 18–24 months + | Attachment behaviors displayed toward several different people, such as siblings, grandparents, or other close family members or caregivers |
Factors Affecting Attachment
A parent or a caregiver who is undergoing their own illness related to labor and delivery or who is experiencing postpartum depression is unable to move toward attachment until first conquering these health-related issues. Unplanned pregnancy, a pregnancy that is a product of sexual assault, or general lack of social support can significantly affect bonding. A drug-exposed infant with a birthing parent who has a history of drug use and may be under other social and economic stressors may have more difficulty making appropriate attachment.
Nursing Interventions That Support Attachment and Bonding
The nurse uses their skills of assessing to determine what attachment behaviors in parents or caregivers and their newborn have occurred. This requires skills beyond those needed for a physical assessment. Nurses rely on their interviewing skills and keen observation of the interactions between the caregivers and their newborn to determine how well attachment and bonding are occurring.
Unexpected findings at birth, such as a genetic or congenital disorder along with subsequent hospitalization of the newborn, can interrupt and hinder attachment between the infant and their caregiver. Nursing interventions to support the attachment process are encouraging skin-to-skin contact when able and advocating for the caregiver to fulfill their role as much as possible in the inpatient setting. Emphasizing common characteristics or individualizing the actions of the newborn helps normalize the child’s behavior for the stressed parent of an inpatient infant. The nurse arranging or encouraging frequent parental visits to the NICU can significantly improve bonding and attachment.
Attachment and the Rest of the Family
In addition to parents, a newborn’s family may consist of siblings, aunts, uncles, cousins, and grandparents. Some of them may welcome their new family member wholeheartedly. Others may be more ambivalent.
Sibling Adjustment
A family is an open system where members can leave or be added at any time. The addition of a new sibling makes for change within the family and a new order for the siblings. The caregiver bringing home a high-risk infant may be caring not only for a fragile newborn but for other children as well. In that case, the caregiver must juggle the many needs of the infant with the needs of their siblings (Volling, 2017).
Temporary separation from the birthing parent or both parents, changes in parental behavior, and the arrival of the new sibling all affect the infant’s siblings. These responses can be positive or negative. Positive responses are more likely when the parents promote sibling acceptance of a new member into the family by including the siblings in preparing for the newborn and continuing to include them in care after the infant arrives home. Positive responses include interest in and concern for the new family member. Negative responses include regression in toileting and sleep. Jealousy of a sibling, sibling rivalry , is common as the new high-risk infant takes up more parental time and energy.
Grandparent Adaptation
Taking on the role of grandparent or even great-grandparent is often anticipated with excitement and happiness. The role of the grandparent is to support their child in becoming a parent. Grandparents can have mixed emotions about accepting a high-risk infant into the family. They may not understand the different responsibilities of their child or healthcare needs of their grandchild. Fear of hurting the infant or uncertainty about the potential outcomes can be scary for new grandparents of a high-risk infant. Including grandparents to the extent that they want to be included is the goal. It is important to acknowledge that parenting and caring for a medically fragile infant is different from the parenting they did in the past.
Increasingly, grandparents are taking on parental roles in providing permanent care for their grandchildren. Including these grandparents early in the care of the high-risk neonate will improve the process of attachment and ease when going home.
Verywell Family provides information for families and has a resource to support new grandparents of premature grandchildren for bonding with the newborn and supporting the baby’s parents.


29.6: Stereochemistry of Cycloadditions
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- Steven Farmer & Kevin M. Shea
- University of Illinois Springfield
Frontier orbital theory can be used to predict if a given cycloaddition will occur with suprafacial or with antarafacial geometry. In a standard Diels-Alder reaction, bonding interactions are created when the electron containing HOMO of the diene donates electrons to the electron vacant LUMO of the other the dienophile. The dienophile has one pi bond, so it will use the pi MOs for a 2 atom system. The dienophile has 2 pi electrons which makes psi 2* its LUMO. The diene has two pi bonds, so it will use the pi MOs for a 4 atom system. The diene has has 4 pi electrons makes psi 2 its HOMO.
These MO diagrams show that the ground-state frontier orbitals of both reactants have terminal lobes with matching signs. The symmetry of these orbitals are such that bond formation will easily occur under thermal conditions with suprafacial geometry as shown below. (Note: The dashed black lines in the figure below represent nodes in the pi molecular orbitals of the diene and dienophile.) The two new sigma bonds, shown as dashed magenta lines below, are formed from constructive overlap of the terminal dienophile orbitals with the terminal orbitals of the diene.
Photochemical [2+2] cycloadditions are excellent reactions for the synthesis of strained products containing 4-membered rings. One of the reaction partners must be conjugated so that it can absorb light and become an excited state molecule. These reactions produce strained 4-membered rings but are not reversible because the products lack conjugation and, thus, can't absorb light to facilitate a cycloreversion.
So, why won't this reaction happen thermally? How does our molecular orbital analysis help us understand the importance of this being a photochemical reaction? First, let's look at the orbital analysis if we tried to do a thermal [2+2] reaction. As shown below, we cannot get suprafacial overlap for both of the 2 pi reactants when trying to combine psi 1 HOMO with psi 2* LUMO. This means that it is not favorable to convert the two reactant pi bonds into two new product pi bonds.
What happens when we shine light on the reaction? Light creates an excited state molecule by promoting an electron in the HOMO to the LUMO, as shown below. This means the excited state HOMO is the ground state LUMO. We need to understand a few key points about photoreactions before doing our molecular orbital analysis. Excited state molecules are very short lived, relaxing back to the ground state very quickly. Therefore, it is practically impossible for two excited state molecules to find each other in a reaction. Instead, reactions occur between one excited state molecule and one ground state molecule. When only one molecule is conjugated, that is the molecule that will form the excited state. If both reactants are conjugated, either can form the excited state. The orbital analysis is shown below. First, we see the orbital picture when a ground state molecule absorbs light to form an excited state. Second, when we analyze the reaction, it is now psi 2* HOMO of the excited state molecule reacting with psi 2* LUMO of the ground state molecule. This gives suprafacial constructive overlap for both orbitals and the 2 reactant pi bonds can be converted into two product sigma bonds.
Cycloaddition reactions can be categorized based on the total number electrons involved in the rearrangement. The [4+2] Diels-Alder reaction involves 6 electrons and takes place using a suprafacial pathway under thermal conditions. The thermal [2+2] cycloaddtion of two alkenes involves 4 electrons and must take place by antarafacial pathway. However, the pathway is reversed for photochemical [2+2] cycloaddtions which take place by a supraficial pathway. These ideas can be generalized using the rules for cycloadditions below.
| Number of Electrons | Thermal | Photochemical |
|---|---|---|
| 4n + 2 | Suprafacial (Allowed) | Antarafacial (Forbidden) |
| 4n | Antarafacial (Forbidden) | Suprafacial (Allowed) |
Exercise \(\PageIndex{1}\)
For the following reactions determine what type of cycloaddition is occurring, is the reaction supra or antarafacial, and would the reaction require thermal or photochemical conditions.
a)
b)
c)
- Answer
-
a) [6+4], suprafacial, thermal
b) [4+2], suprafacial, thermal
c) [2+2], antarafacial, photochemical


1.5.10: Chemical Potentials- Solutions- Henry's Law
A given aqueous solutions at temperature \(\mathrm{T}\) contains a simple solute \(j\), molality \(\mathrm{m}_{j}\). Experiment shows that at equilibrium the partial pressure \(\mathrm{p}_{j}\) is close to a linear function of molality \(\mathrm{m}_{j}\), the constant of proportionality being the Henry’s Law constant for this particular solute in a defined solvent; equation(a).
\[\mathrm{p}_{\mathrm{j}} \cong \mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
By definition \(\mathrm{m}^{0}=1.0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\). Experiment shows that as a given real solution becomes more dilute so the relationship given in (a) can be written as an equation. The relationship in (a) is rewritten as an equation to describe the properties of a solution having thermodynamic properties which are ideal.
\[p_{j}(\mathrm{id})=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
In other words \(\mathrm{H}_{j}\) is the partial pressure of volatile solute in a solution having thermodynamic properties which are ideal and where the molality of the solute equals \(1.0 \mathrm{~mol kg}^{-1}\).
For a real solution at equilibrium and at temperature \(\mathrm{T}\), the partial pressure \(\mathrm{p}_{\mathrm{j}}(\text { real })\) is related to molality mj using equation (c) where \(\gamma_{j}\) is the activity coefficient describing the properties of solute \(j\) in solution
\[\mathrm{p}_{\mathrm{j}}(\text { real })=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \, \gamma_{\mathrm{j}} \nonumber \]
\[\text { By definition, at all T and } p \lim \operatorname{it}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1 \nonumber \]
As \(\mathrm{m}_{j}\) decreases so \(\mathrm{p}_{\mathrm{j}}(\text { real })\) approaches \(\mathrm{p}_{\mathrm{j}}(\mathrm{id})\).
Henry’s Law forms the basis of equations which are used to related the chemical potential of solute \(j\), \(\mu_{j}\) to the composition of a solution.


5.1: The order of Integers and Primitive Roots
In this section, we study the order of an integer modulo \(n\), where \(n\) is positive. We also define primitive roots and related results. Euler’s theorem in Chapter 4 states that if a positive integer \(a\) is relatively prime to \(n\), then \(a^{\phi(n)}\equiv 1 (mod \ n)\). Thus by the well ordering principle, there is a least positive integer \(x\) that satisfies this congruence \(a^{x}\equiv 1 (mod \ n)\).
Let \((a,b)=1\). The smallest positive integer \(x\) such that \(a^{x} \equiv 1(mod \ b)\) is called the order of \(a\) modulo \(b\). We denote the order of \(a\) modulo \(b\) by \(ord_ba\).
\(ord_72=3\) since \(2^3\equiv 1(mod \ 7)\) while \(2^1\equiv 2(mod \ 7)\) and \(2^2\equiv 4(mod \ 7)\).
To find all integers \(x\) such that \(a^x\equiv 1(mod \ b)\), we need the following theorem.
If \((a,b)=1\) with \(b>0\), then the positive integer \(x\) is a solution of the congruence \(a^x\equiv 1(mod \ b)\) if and only if \(ord_ba\mid x\).
Having \(ord_ba\mid x\), then we have that \(x=k.ord_ba\) for some positive integer \(k\). Thus \[a^x=a^{kord_ba}=(a^{ord_ba})^k\equiv 1(mod \ b).\] Now if \(a^x\equiv 1(mod \ b)\), we use the division algorithm to write \[x=q ord_ba+r, \ \ \ 0\leq r<ord_ba.\] Thus we see that \[a^x\equiv a^{qord_ba+r}\equiv (a^{ord_ba})^qa^r\equiv a^r (mod \ b).\] Now since \(a^x\equiv 1(mod \ b)\),we have \(a^r\equiv 1(mod \ b)\). Since \(ord_ba\), we get \(r=0\). Thus \(x=q.ord_ba\) and hence \(ord_ba\mid x\).
Since \(ord_72=3\), then \(2^{15}\equiv 1(mod \ 7)\) while 10 is not a solution for \(2^x\equiv 1 (mod \ 7)\).
If \((a,b)=1\) with \(b>0\), then \[a^i\equiv a^j(mod \ b)\] where \(i\) and \(j\) are nonnegative integers, if and only if \[i\equiv j(mod \ ord_ba)\]
Suppose that \[i\equiv j(mod \ ord_ba)\ \ \mbox{and}\ \ 0\leq j\leq i.\] Then we have \(i-j=k.ord_ba\), where \(k\) is a positive integer. Hence \[a^i=a^{j+k.ord_ba}=a^j(a^{ord_ba})^k\equiv a^j (mod \ b).\] Assume now that \(a^i\equiv a^j(mod \ b)\) with \(i\geq j\). Thus we have \[a^i\equiv a^ja^{i-j}\equiv a^j(mod \ b)\] Since \((a,b)=1\), we have \((a^j,b)=1\) and thus by Theorem 22, we get \[a^{i-j}\equiv 1(mod \ b).\] By theorem 54, we get that \(ord_ba \mid (i-j)\) and hence \(i\equiv j(mod \ b)\).
We introduce now primitive roots and discuss their properties. We are interested in integers whose order modulo another integer is \(\phi(b)\). In one of the exercises, one is asked to prove that if \(a\)and \(b\) are relatively prime then \(ord_ba \mid \phi(b)\).
If \((r,m)=1\) with \(m>0\) and if \(ord_mr=\phi(m)\) then \(r\) is called a primitive root modulo \(m\).
Notice that \(\phi(7)=6\) hence \(2\) is not a primitive root modulo \(7\). While \(ord_73=6\) and thus \(3\) is a primitive root modulo \(7\).
If \((r,m)=1\) with \(m>0\) and if \(r\) is a primitive root modulo \(n\), then the integers \(\{r^1,r^2,...r^{\phi(m)}\}\) form a reduced residue set modulo \(m\).
To prove that the set \(\{r^1,r^2,...r^{\phi(m)}\}\) form a reduced residue set modulo \(m\) we need to show that every two of them are relatively prime and that no two of them are congruent modulo \(m\). Since \((r,m)=1\), it follows that \((r^n,m)=1\) for all positive integers \(n\). Hence all the powers of \(r\) are relatively prime to \(m\). To show that no two powers in the above set are equivalent modulo \(m\), assume that \[r^i\equiv r^j(mod \ m).\] By Theorem 55, we see that \[i\equiv j(mod \ ord_m\phi(m)).\] Notice that \(1\leq i,j\leq \phi(m)\) and hence \(i=j\).
If \(ord_ma=t\) and if \(u\) is a positive integer, then \[ord_m(a^u)=t/(t,u).\]
 Notice that \((t_1,u_1)=1.\)
Because \(t_1=t/(t,u)\), we want to show that \(ord_m(a^u)=t_1\). To do this, we will show that \((a^{u})^{t_1}\equiv 1(mod \ m)\) and that if \((a^u)^v\equiv 1(mod \ m)\), then \(t_1\mid v\). First note that \[(a^u)^{t_1}=(a^{u_1w})^{(t/w)}=(a^t)^{u_1}\equiv 1(mod \ m).\] Hence by Theorem 54, we have \(v\mid t_1\). Now on the other hand, since \[(a^u)^v=a^{uv}\equiv 1(mod \ m),\] we know that \(t\mid uv\). Hence \(t_1w\mid u_1wv\) and hence \(t_1\mid u_1v\). Because \((t_1,u_1)=1\), we see that \(t_1\mid v\). Since \(v\mid t_1\) and \(t_1\mid v\), we conclude that \(v=t_1=t/w=t/(t,u)\).
We see that \(ord_73^4=6/(6,4)\) since \(ord_73=6\).
Let \(r\) be a primitive root modulo \(m\), where \(m\) is a positive integer, \(m>1\). Then \(r^u\) is a primitive root modulo \(m\) if and only if \((u,\phi(m))=1\).
By Theorem 57, we see that \[ord_mr^u=ord_mr/(u,ord_mr)=\phi(m)/(u,\phi(m)).\] Thus \(ord_mr^u=\phi(m)\) and \(r^u\) is a primitive root if and only if \((u,\phi(m))=1\).
The above corollary leads to the following theorem
If the positive integer \(m\) has a primitive root, then it has a total of \(\phi(\phi(m))\) incongruent primitive roots.
Let \(r\) be a primitive root modulo \(m\). By Theorem 56, we see that \(\{r^1,r^2,...,r^{\phi(m)}\}\) form a reduced residue system modulo \(n\). By Corollary 1, it is known that \(r^u\) is a primitive root modulo \(m\) if and only if \((u,\phi(m))=1\). Thus we have exactly \(\phi(\phi(m))\) such integers \(u\) that are relatively prime to \(\phi(m)\) and hence there are exactly \(\phi(\phi(m))\) primitive roots modulo \(m\).
Exercises
- Determine \(ord_{13}10\) .
- Determine \(ord_{11}3.\)
- Show that 5 is a primitive root of 6.
- Show that if \(\bar{a}\) is an inverse of \(a\) modulo \(n\), then \(ord_na=ord_n\bar{a}\).
- Show that if \(n\) is a positive integer, and \(a\) and \(b\) are integers relatively prime to \(n\) such that \((ord_na,ord_nb)=1\), then \(ord_n(ab)=ord_na.ord_nb\).
- Show that if \(a\) is an integer relatively prime to the positive integer \(m\) and \(ord_ma=st\), then \(ord_ma^t=s\).
- Show that if \(a\) and \(n\) are relatively prime with \(n>0\), then \(ord_na \mid \phi(n)\).
Contributors and Attributions
-
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


2.1: Traversing Circles
- How does a point traversing a circle naturally generate a function?
- What are some important properties that characterize a function generated by a point traversing a circle?
- How does a circular function change in ways that are different from linear and quadratic functions?
Certain naturally occurring phenomena eventually repeat themselves, especially when the phenomenon is somehow connected to a circle. For example, suppose that you are taking a ride on a ferris wheel and we consider your height, \(h\text{,}\) above the ground and how your height changes in tandem with the distance, \(d\text{,}\) that you have traveled around the wheel. In Figure \(\PageIndex{1}\) we see a snapshot of this situation, which is available as a full animation 1 at http://gvsu.edu/s/0Dt .
In the context of the ferris wheel pictured in Figure \(\PageIndex{1}\) , assume that the height, \(h\text{,}\) of the moving point (the cab in which you are riding), and the distance, \(d\text{,}\) that the point has traveled around the circumference of the ferris wheel are both measured in meters.
Further, assume that the circumference of the ferris wheel is \(150\) meters. In addition, suppose that after getting in your cab at the lowest point on the wheel, you traverse the full circle several times.
- Recall that the circumference, \(C\text{,}\) of a circle is connected to the circle's radius, \(r\text{,}\) by the formula \(C = 2\pi r\text{.}\) What is the radius of the ferris wheel? How high is the highest point on the ferris wheel?
- How high is the cab after it has traveled \(1/4\) of the circumference of the circle?
- How much distance along the circle has the cab traversed at the moment it first reaches a height of \(\frac{150}{\pi} \approx 47.75\) meters?
- Can \(h\) be thought of as a function of \(d\text{?}\) Why or why not?
- Can \(d\) be thought of as a function of \(h\text{?}\) Why or why not?
- Why do you think the curve shown at right in Figure 2.1.1 has the shape that it does? Write several sentences to explain.
Circular Functions
The natural phenomenon of a point moving around a circle leads to interesting relationships. For easier arithmetic, let's consider a point traversing a circle of circumference \(24\) and examine how the point's height, \(h\text{,}\) changes as the distance traversed, \(d\text{,}\) changes. Note particularly that each time the point traverses \(\frac{1}{8}\) of the circumference of the circle, it travels a distance of \(24 \cdot \frac{1}{8} = 3\) units, as seen in Figure \(\PageIndex{2}\) where each noted point lies \(3\) additional units along the circle beyond the preceding one.
Note that we know the exact heights of certain points. Since the circle has circumference \(C = 24\text{,}\) we know that \(24 = 2\pi r\) and therefore \(r = \frac{12}{\pi} \approx 3.82\text{.}\) Hence, the point where \(d = 6\) (located \(1/4\) of the way along the circle) is at a height of \(h = \frac{12}{\pi} \approx 3.82\text{.}\) Doubling this value, the point where \(d = 12\) has height \(h = \frac{24}{\pi} \approx 7.64\text{.}\) Other heights, such as those that correspond to \(d = 3\) and \(d = 15\) (identified on the figure by the green line segments) are not obvious from the circle's radius, but can be estimated from the grid in Figure \(\PageIndex{2}\) as \(h \approx 1.1\) (for \(d = 3\)) and \(h \approx 6.5\) (for \(d = 15\)). Using all of these observations along with the symmetry of the circle, we can determine the other entries in Table \(\PageIndex{3}\)
| \(d\) | \(0\) | \(3\) | \(6\) | \(9\) | \(12\) | \(15\) | \(18\) | \(21\) | \(24\) |
| \(h\) | \(0\) | \(1.1\) | \(3.82\) | \(6.5\) | \(7.64\) | \(6.5\) | \(3.82\) | \(1.1\) | \(0\) |
Moreover, if we now let the point continue traversing the circle, we observe that the \(d\)-values will increase accordingly, but the \(h\)-values will repeat according to the already-established pattern, resulting in the data in Table \(\PageIndex{4}\)
| \(d\) | \(24\) | \(27\) | \(30\) | \(33\) | \(36\) | \(39\) | \(42\) | \(45\) | \(48\) |
| \(h\) | \(0\) | \(1.1\) | \(3.82\) | \(6.5\) | \(7.64\) | \(6.5\) | \(3.82\) | \(1.1\) | \(0\) |
It is apparent that each point on the circle corresponds to one and only one height, and thus we can view the height of a point as a function of the distance the point has traversed around the circle, say \(h = f(d)\text{.}\) Using the data from the two tables and connecting the points in an intuitive way, we get the graph shown in Figure \(\PageIndex{5}\)
The function \(h = f(d)\) we have been discussing is an example of what we will call a circular function . Indeed, it is apparent that if we
- take any circle in the plane,
- choose a starting location for a point on the circle,
- let the point traverse the circle continuously,
- and track the height of the point as it traverses the circle,
the height of the point is a function of distance traversed and the resulting graph will have the same basic shape as the curve shown in Figure 2.1.5. It also turns out that if we track the location of the \(x\)-coordinate of the point on the circle, the \(x\)-coordinate is also a function of distance traversed and its curve has a similar shape to the graph of the height of the point (the \(y\)-coordinate). Both of these functions are circular functions because they are generated by motion around a circle.
Consider the circle pictured in Figure \(\PageIndex{6}\) that is centered at the point \((2,2)\) and that has circumference \(8\text{.}\) Assume that we track the \(y\)-coordinate (that is, the height, \(h\)) of a point that is traversing the circle counterclockwise and that it starts at \(P_0\) as pictured.
- How far along the circle is the point \(P_1\) from \(P_0\text{?}\) Why?
- Label the subsequent points in the figure \(P_2\text{,}\) \(P_3\text{,}\) \(\ldots\) as we move counterclockwise around the circle. What is the exact \(y\)-coordinate of the point \(P_2\text{?}\) of \(P_4\text{?}\) Why?
- Determine the \(y\)-coordinates of the remaining points on the circle (exactly where possible, otherwise approximately) and hence complete the entries in Table \(\PageIndex{8}\) that track the height, \(h\text{,}\) of the point traversing the circle as a function of distance traveled, \(d\text{.}\) Note that the \(d\)-values in the table correspond to the point traversing the circle more than once.
| \(d\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) |
| \(h\) | \(2\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
4. By plotting the points in Table \(\PageIndex{8}\) and connecting them in an intuitive way, sketch a graph of \(h\) as a function of \(d\) on the axes provided in Figure \(\PageIndex{7}\) over the interval \(0 \le d \le 16\text{.}\) Clearly label the scale of your axes and the coordinates of several important points on the curve.
5. What is similar about your graph in comparison to the one in Figure \(\PageIndex{5}\) ? What is different?
6. What will be the value of \(h\) when \(d = 51\text{?}\) How about when \(d = 102\text{?}\)
Properties of Circular Functions
Every circular function has several important features that are connected to the circle that defines the function. For the discussion that follows, we focus on circular functions that result from tracking the \(y\)-coordinate of a point traversing counterclockwise a circle of radius \(a\) centered at the point \((k,m)\text{.}\) Further, we will denote the circumference of the circle by the letter \(p\text{.}\)
We assume that the point traversing the circle starts at \(P\) in Figure \(\PageIndex{9}\) Its height is initially \(y = m + a\text{,}\) and then its height decreases to \(y = m\) as we traverse to \(Q\text{.}\) Continuing, the point's height falls to \(y = m - a\) at \(R\text{,}\) and then rises back to \(y = m\) at \(S\text{,}\) and eventually back up to \(y = m+a\) at the top of the circle. If we plot these heights continuously as a function of distance, \(d\text{,}\) traversed around the circle, we get the curve shown at right in Figure \(\PageIndex{10}\) This curve has several important features for which we introduce important terminology.
The midline of a circular function is the horizontal line \(y = m\) for which half the curve lies above the line and half the curve lies below. If the circular function results from tracking the \(y\)-coordinate of a point traversing a circle, \(y = m\) corresponds to the \(y\)-coordinate of the center of the circle. In addition, the amplitude of a circular function is the maximum deviation of the curve from the midline. Note particularly that the value of the amplitude, \(a\text{,}\) corresponds to the radius of the circle that generates the curve.
Because we can traverse the circle in either direction and for as far as we wish, the domain of any circular function is the set of all real numbers. From our observations about the midline and amplitude, it follows that the range of a circular function with midline \(y = m\) and amplitude \(a\) is the interval \([m-a,m+a]\text{.}\)
Finally, we introduce the formal definition of a periodic function.
Let \(f\) be a function whose domain and codomain are each the set of all real numbers. We say that \(f\) is periodic provided that there exists a real number \(k\) such that \(f(x+k) = f(x)\) for every possible choice of \(x\text{.}\) The smallest value \(p\) for which \(f(x+p) = f(x)\) for every choice of \(x\) is called the period of \(f\text{.}\)
For a circular function, the period is always the circumference of the circle that generates the curve. In Figure \(\PageIndex{10}\) , we see how the curve has completed one full cycle of behavior every \(p\) units, regardless of where we start on the curve.
Circular functions arise as models for important phenomena in the world around us, such as in a harmonic oscillator . Consider a mass attached to a spring where the mass sits on a frictionless surface. After setting the mass in motion by stretching or compressing the spring, the mass will oscillate indefinitely back and forth, and its distance from a fixed point on the surface turns out to be given by a circular function.
A weight is placed on a frictionless table next to a wall and attached to a spring that is fixed to the wall. From its natural position of rest, the weight is imparted an initial velocity that sets it in motion. The weight then oscillates back and forth, and we can measure its distance, \(h = f(t)\) (in inches) from the wall at any given time, \(t\) (in seconds). A graph of \(f\) and a table of select values are given below.
| \(t\) | \(f(t)\) |
| \(0.25\) | \(6.807\) |
| \(0.5\) | \(4.464\) |
| \(0.75\) | \(3.381\) |
| \(1\) | \(3.000\) |
| \(1.25\) | \(3.381\) |
| \(1.5\) | \(4.464\) |
| \(1.75\) | \(6.087\) |
| \(2\) | \(8.000\) |
| \(t\) | \(f(t)\) |
| \(2.25\) | \(9.913\) |
| \(2.5\) | \(11.536\) |
| \(2.75\) | \(12.619\) |
| \(3\) | \(13.000\) |
| \(3.25\) | \(12.619\) |
| \(3.5\) | \(11.536\) |
| \(3.75\) | \(9.913\) |
| \(4\) | \(8.000\) |
- Determine the period \(p\text{,}\) midline \(y = m\text{,}\) and amplitude \(a\) of the function \(f\text{.}\)
- What is the furthest distance the weight is displaced from the wall? What is the least distance the weight is displaced from the wall? What is the range of \(f\text{?}\)
- Determine the average rate of change of \(f\) on the intervals \([4,4.25]\) and \([4.75,5]\text{.}\) Write one careful sentence to explain the meaning of each (including units). In addition, write a sentence to compare the two different values you find and what they together say about the motion of the weight.
- Based on the periodicity of the function, what is the value of \(f(6.75)\text{?}\) of \(f(11.25)\text{?}\)
The average rate of change of a circular function
Just as there are important trends in the values of a circular function, there are also interesting patterns in the average rate of change of the function. These patterns are closely tied to the geometry of the circle.
For the next part of our discussion, we consider a circle of radius \(1\) centered at \((0,0)\text{,}\) and consider a point that travels a distance \(d\) counterclockwise around the circle with its starting point viewed as \((1,0)\text{.}\) We use this circle to generate the circular function \(h = f(d)\) that tracks the height of the point at the moment the point has traversed \(d\) units around the circle from \((1,0)\text{.}\) Let's consider the average rate of change of \(f\) on several intervals that are connected to certain fractions of the circumference.
Remembering that \(h\) is a function of distance traversed along the circle, it follows that the average rate of change of \(h\) on any interval of distance between two points \(P\) and \(Q\) on the circle is given by
where both quantities are measured from point \(P\) to point \(Q\text{.}\)
First, in Figure \(\PageIndex{12}\) , we consider points \(P\text{,}\) \(Q\text{,}\) and \(R\) where \(Q\) results from traversing \(1/8\) of the circumference from \(P\text{,}\) and \(R\) \(1/8\) of the circumference from \(Q\text{.}\) In particular, we note that the distance \(d_1\) along the circle from \(P\) to \(Q\) is the same as the distance \(d_2\) along the circle from \(Q\) to \(R\text{,}\) and thus \(d_1 = d_2\text{.}\) At the same time, it is apparent from the geometry of the circle that the change in height \(h_1\) from \(P\) to \(Q\) is greater than the change in height \(h_2\) from \(Q\) to \(R\text{,}\) so \(h_1 \gt h_2\text{.}\) Thus, we can say that
The differences in certain average rates of change appear to become more extreme if we consider shorter arcs along the circle. Next we consider traveling \(1/20\) of the circumference along the circle. In Figure \(\PageIndex{13}\) , points \(P\) and \(Q\) lie \(1/20\) of the circumference apart, as do \(R\) and \(S\text{,}\) so here \(d_1 = d_5\text{.}\) In this situation, it is the case that \(h_1 \gt h_5\) for the same reasons as above, but we can say even more. From the green triangle in Figure \(\PageIndex{13}\) , we see that \(h_1 \approx d_1\) (while \(h_1 \lt d_1\)), so that \(AV_{[P,Q]} = \frac{h_1}{d_1} \approx 1\text{.}\) At the same time, in the magenta triangle in the figure we see that \(h_5\) is very small, especially in comparison to \(d_5\text{,}\) and thus \(AV_{[R,S]} = \frac{h_5}{d_5} \approx 0\text{.}\) Hence, in Figure \(\PageIndex{13}\) ,
This information tells us that a circular function appears to change most rapidly for points near its midline and to change least rapidly for points near its highest and lowest values.
We can study the average rate of change not only on the circle itself, but also on a graph such as Figure \(\PageIndex{10}\) , and thus make conclusions about where the function is increasing, decreasing, concave up, and concave down.
Consider the same setting as Activity 2.1.3: a weight oscillates back and forth on a frictionless table with distance from the wall given by, \(h = f(t)\) (in inches) at any given time, \(t\) (in seconds). A graph of \(f\) and a table of select values are given below.
| \(t\) | \(f(t)\) |
| \(0.25\) | \(6.807\) |
| \(0.5\) | \(4.464\) |
| \(0.75\) | \(3.381\) |
| \(1\) | \(3.000\) |
| \(1.25\) | \(3.381\) |
| \(1.5\) | \(4.464\) |
| \(1.75\) | \(6.087\) |
| \(2\) | \(8.000\) |
| \(t\) | \(f(t)\) |
| \(2.25\) | \(9.913\) |
| \(2.5\) | \(11.536\) |
| \(2.75\) | \(12.619\) |
| \(3\) | \(13.000\) |
| \(3.25\) | \(12.619\) |
| \(3.5\) | \(11.536\) |
| \(3.75\) | \(9.913\) |
| \(4\) | \(8.000\) |
- Determine \(AV_{[2,2.25]}\text{,}\) \(AV_{[2.25,2.5]}\text{,}\) \(AV_{[2.5,2.75]}\text{,}\) and \(AV_{[2.75,3]}\text{.}\) What do these four values tell us about how the weight is moving on the interval \([2,3]\text{?}\)
- Give an example of an interval of length \(0.25\) units on which \(f\) has its most negative average rate of change. Justify your choice.
- Give an example of the longest interval you can find on which \(f\) is decreasing.
- Give an example of an interval on which \(f\) is concave up. 2
- On an interval where \(f\) is both decreasing and concave down, what does this tell us about how the weight is moving on that interval? For instance, is the weight moving toward or away from the wall? is it speeding up or slowing down?
- What general conclusions can you make about the average rate of change of a circular function on intervals near its highest or lowest points? about its average rate of change on intervals near the function's midline?
Summary
- When a point traverses a circle, a corresponding function can be generated by tracking the height of the point as it moves around the circle, where height is viewed as a function of distance traveled around the circle. We call such a function a circular function . An image that shows how a circular function's graph is generated from the circle can be seen in Figure 2.1.10.
- Circular functions have several standard features. The function has a midline that is the line for which half the points on the curve lie above the line and half the points on the curve lie below. A circular function's amplitude is the maximum deviation of the function value from the midline; the amplitude corresponds to the radius of the circle that generates the function. Circular functions also repeat themselves, and we call the smallest value of \(p\) for which \(f(x+p) = f(x)\) for all \(x\) the period of the function. The period of a circular function corresponds to the circumference of the circle that generates the function.
- Non-constant linear functions are either always increasing or always decreasing; quadratic functions are either always concave up or always concave down. Circular functions are sometimes increasing and sometimes decreasing, plus sometimes concave up and sometimes concave down. These behaviors are closely tied to the geometry of the circle.
Exercises
Consider the circle pictured in Figure \(\PageIndex{14}\) that is centered at the point \((2,2)\) and that has circumference \(8\text{.}\) Suppose that we track the \(x\)-coordinate (that is, the horizontal location, which we will call \(k\)) of a point that is traversing the circle counterclockwise and that it starts at \(P_0\) as pictured.
Recall that in Activity \(\PageIndex{2}\) we identified the exact and approximate vertical coordinates of all \(8\) noted points on the circle. In addition, recall that the radius of the circle is \(r = \frac{8}{2\pi} \approx 1.2732\text{.}\)
- What is the exact horizontal coordinate of \(P_0\text{?}\) Why?
- Complete the entries in Table \(\PageIndex{16}\) that track the horizontal location, \(k\text{,}\) of the point traversing the circle as a function of distance traveled, \(d\text{.}\)
| \(d\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) |
| \(k\) | \(0.73\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
3. By plotting the points in Table \(\PageIndex{16}\) and connecting them in an intuitive way, sketch a graph of \(k\) as a function of \(d\) on the axes provided in Figure \(\PageIndex{15}\) over the interval \(0 \le d \le 16\text{.}\) Clearly label the scale of your axes and the coordinates of several important points on the curve.
4. What is similar about your graph in comparison to the one in Figure \(\PageIndex{7}\) ? What is different?
5. What will be the value of \(k\) when \(d = 51\text{?}\) How about when \(d = 102\text{?}\)
Two circular functions, \(f\) and \(g\text{,}\) are generated by tracking the \(y\)-coordinate of a point traversing two different circles. The resulting graphs are shown in Figure \(\PageIndex{17}\) and Figure \(\PageIndex{18}\) Assuming the horizontal scale matches the vertical scale, answer the following questions for each of the functions \(f\) and \(g\text{.}\)
- Assume that the circle used to generate the circular function is centered at the point \((0,m)\) and has radius \(r\text{.}\) What are the numerical values of \(m\) and \(r\text{?}\) Why?
- What are the coordinates of the location on the circle at which the point begins its traverse? Said differently, what point on the circle corresponds to \(t = 0\) on the function's graph?
- What is the period of the function? How is this connected to the circle and to the scale on the horizontal axes on which the function is graphed?
- How would the graph look if the circle's radius was \(1\) unit larger? \(1\) unit smaller?
A person goes for a ride on a ferris wheel. They enter one of the cars at the lowest possible point on the wheel from a platform \(7\) feet off the ground. When they are at the very top of the wheel, they are \(92\) feet off the ground. Let \(h\) represent the height of the car (in feet) and \(d\) (in feet) the distance the car has traveled along the wheel's circumference from its starting location at the bottom of the wheel. We'll use the notation \(h = f(d)\) for how height is a function of distance traveled.
- How high above the ground is the center of the ferris wheel?
- How far does the car travel in one complete trip around the wheel?
- For the circular function \(h = f(d)\text{,}\) what is its amplitude? midline? period?
- Sketch an accurate graph of \(h\) through at least two full periods. Clearly label the scale on the horizontal and vertical axes along with several important points.
<1> Used with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved.
<2> Recall that a function is concave up on an interval provided that throughout the interval, the curve bends upward, similar to a parabola that opens up.


 Was it an insult? A compliment? We have been taught that throwing like a girl is the equivalent of throwing poorly. We use a variety of channels of communication (language, books, tv, clothing, etc.) to teach children what it means to be a “girl” or a “boy”. We limit these identities to separate categories we are not usually supposed to mix. We are taught that men are supposed to be more athletic than women, and even play in different leagues! In almost every professional sport like football, baseball, and basketball the men’s league is seen as more competitive and more popular. The company Always decided to examine the phrase “ like a girl ” and how children of different ages would respond! The results were not what you would expect!
So what happens when a girl is able to throw a 70 mile per hour fastball and win The World Series for her team? Ideas about gender are continually being challenged. Mo’ne Davis was the first girl to pitch a shutout in the Little League World Series in August 2014 and is showing everyone what it means to throw like a girl. Known as America’s favorite pastime, baseball used to be reserved for men only. Currently, Davis is being recognized because of her rare talent, but also because of her gender (Wallace). With Davis as a role model, we hope to see many more examples of transformations of traditional gender roles! For more information on Mo’ne, check out this link ! This example highlights one of the key characteristics of gender—-that it is fluid. Gender roles of a given culture are always changing.
Like in sports, people of all genders are taking on new roles in all different ways! This picture depicts females on the field during a competitive game of lacrosse at Humboldt State University in Arcata, California.
In this chapter, we want to look at the ways in which gender has been constructed in our society and the ways in which we communicate about the idea of gender.
Gender communication is a specialization of the communication field that focuses on the ways we, as gendered beings, communicate. Gender research might look at roles for people of different genders in academia, sports, media, or politics. For example, research in this area could examine the similarities and differences in the conversations that take place in the comment section of a Youtube video created by Bethany Mota verses one created by Philip DeFranco. Researchers could also look at how people of different genders have been represented throughout history. Gender communication is also a field that strives to change the way we talk about people, in order to make a more empathetic and safe space for our entire community.  Now we see the LGBTQIA (lesbian, gay, bisexual, trans, queer/questioning, intersex, asexual) community has reclaimed the word queer to mean any person who is not straight. It is now a self proclamation and one that can be empowering for many people.
In this chapter, we want to make a distinction between sex and gender before providing an overview of this specialization’s areas of research, main theories and theorists, and highlights from research findings about feminine and masculine communication styles. While we are taking a communication lens to the study of gender, we need to acknowledge the contributions made by other academic disciplines such as women’s studies, linguistics, and psychology (Stephen, 2000).
As with other specializations in communication, definitions of gender abound (Gamble & Gamble; Gilbert; Howard & Hollander; Lorber; Vannoy). Ivy and Backlund define gender communication as, ‘ “communication about and between men and women”’ (4). Central to this definition are the terms about and between, and men and women. About addresses the attention this specialization pays to how the sexes are “discussed, referred to, or depicted, both verbally and nonverbally.”  We find this problematic because it limits the discussion about gender to only men and women. For our purposes, we will be adapting the Ivy and Backlund definition and instead using the definition: communication about and between people of all genders. This new definition is more inclusive of the large number of gender identities that are present in our community. For example gender queer, transgender, and a-gender. We will discuss and define some of these identities later in the chapter, for a more in-depth exploration of these identities, check out this article from the Huffington Post .
In our society, we use the gendered terms women and men instead of male and female. What’s the difference between these two sets of terms? One pair refers to the biological categories of male and female. The other pair, men and women, refers to what are now generally regarded as socially constructed concepts that convey the cultural ideals or values of masculinity and femininity. For our purposes, gender is, “the social construction of masculinity or femininity as it aligns with designated sex at birth in a specific culture and time period. Gender identity claims individuality that may or may not be expressed outwardly, and may or may not correspond to one’s sexual anatomy” (Pettitt). This definition is important because it discusses the separation between sex and gender as well as the idea that gender is socially constructed.
This basic difference is important, but it’s most important that you know something else about these two sets of terms. One set has fixed meaning and the other set maintains fluid or dynamic meaning. Because they refer to biological distinctions, the terms male and female are essentially fixed. That is, their meanings are generally unchanging (as concepts if not in reality, since we do live in an age when it’s medically possible to change sexes). Conversely, because they are social constructions, the meanings of the gendered terms masculine and feminine are dynamic or fluid. Why? Because their meanings are open to interpretation: Different people give them different meanings. Sometimes, even the same person might interpret these terms differently over time. For example, as a teenager a girl may portray her femininity by wearing make-up. Eventually, she may decide to forego this traditional display of femininity because her sense of herself as a woman may no longer need the validation that a socially prescribed behavior, such as wearing make-up, provides. We use the terms fluid and dynamic to describe the social construction of gender because they will change based on the time, place, and culture a person lives in. For example, did you know that high heels were first invented for men to make them look taller? These days, if a man wears high heels, he would be described as “feminine.” This is an example of how our ideas of gender can change over time.
Contributions and Affiliations
- Survey of Communication Study. Authored by : Scott T Paynton and Linda K Hahn. Provided by : Humboldt State University. Located at : en.wikibooks.org/wiki/Survey_of_Communication_Study/Preface. License : CC BY-SA: Attribution-ShareAlike


17.4: Analyzing Secondary Dominants
17.4 Analyzing Secondary Dominants
When you encounter a chord with a chromaticism and suspect it is a secondary dominant, use the following process.
-
Stack the chord in thirds to determine the root and quality. If the chord quality is major (if a triad) or a major–minor seventh chord, go on to step 2. If the chord quality is not major or major–minor seventh, the chord is not a secondary dominant.
The chord in question is an F♯ major triad in first inversion.
-
Determine the note that would be a perfect 5th below the
root
of the chord you are analyzing. If this note would be the root of a diatonic chord, the chord you are analyzing is a secondary dominant.
Since B is 5^ , the F♯ major chord in first inversion is tonicizing VV. Therefore the chord is VVV6/V.